Accurate analysis of structural differences and commonalities is of fundamental importance for understanding the structure, function, and evolution of biological macromolecules. For the past 40 years, structural analysis methods have relied on the biophysically unrealistic and restrictive least-squares criterion to find optimal superpositions. By developing probabilistic models of structural change that can take advantage of powerful maximum likelihood (ML) and Bayesian techniques, this proposed work will greatly expand our abilities to accurately superposition, align, and analyze structural conformations. The specific aims of this work are (1) to develop Bayesian models and theory for superpositioning structural conformation, (2) develop ML and Bayesian models and theory for multiple structure-based alignment, and (3) develop and distribute computational tools that implement this ML and Bayesian structural analysis. ML and Bayesian structural analysis will provide many distinct advantages over current least-squares and other ad hoc methods, including (1) straight- forward estimates of the uncertainty in the solutions of estimated parameters, (2) elegant handling of uncertainty in structural data, (3) natural incorporation of disparate types of prior structural and molecular information, (4) easy examination of complex models of structural change and evolution, (5) rigorous testing of complex structural hypotheses, and (6) natural handling of missing structural data. While we concentrate specifically on the conformations of macromolecules, the methods proposed herein have broad mathematical generality and will impact not only molecular structural biology but also an unusually wide range of scientific fields, including any that compare the shapes and conformations of objects. The results developed from this work will be applicable to any entity that can be represented as a set of Cartesian points in a multi-dimensional space, whether the particular structures under study are proteins, skulls, MRI scans, geological strata, or even psychological profiles of human individuals. PUBLIC HEALTH RELEVANCE: Measuring, analyzing, and comparing the shapes and conformations of the structures of objects is of fundamental importance in many diverse scientific fields. Our particular focus is the development of likelihood and Bayesian methods for the comparison and analysis of multiple three-dimensional macromolecules. While we concentrate specifically on the conformations of macromolecules, the methods proposed herein will be generally applicable to any entity that can be represented as a set of Cartesian points in a multi-dimensional space, whether the particular structures under study are proteins, skulls, MRI scans, geological strata, or even psychological profiles of human individuals.